Interior points

  • Thread starter omri3012
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  • #1
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Hallo,

My teacher wrote that:

"The set has no interior points, and neither does its complement, R\Q" where R refers real

numbers and Q is the rationals numbers.

why can't i find an iterior point?

thanks,

Omri
 

Answers and Replies

  • #2
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Let q be an arbitrary rational number. Does there exist a neighborhood of q that is a subset of Q?
 
  • #3
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How about the fact that I can squeeze a real number between any two arbitrary points of Q?
 
  • #4
HallsofIvy
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How about the fact that I can squeeze a real number between any two arbitrary points of Q?
Irrelevant. Do you mean an irrational number? Now that would be relevant.
 
  • #5
HallsofIvy
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Hallo,

My teacher wrote that:

"The set has no interior points, and neither does its complement, R\Q" where R refers real

numbers and Q is the rationals numbers.

why can't i find an iterior point?

thanks,

Omri
So "the set" is Q. For p to be an interior point of Q, there must exist an interval around p, \(\displaystyle (p-\delta, p+\delta)[/quote] consisting entirely of rational numbers. For p to be an interior point of R\Q, the set of irrational numbers, there must exist an interval [itex](p- \delta, p+ \delta)[/itex]] consisting entirely of irrational numbers. There is NO interval of real numbers consisting entirely of rational number or entirely of irrational numbers.\)
 
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  • #6
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That was what I said anyway, but of course a real is not necessarily rational part got lost along the way... Sorry for that.
 
  • #7
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thank you for your comments,

I'm sorry but the statement (as i guess you already assume) was:

"The set Q has no interior points, and neither does its complement, R\Q"

thanks

Omri
 
  • #8
HallsofIvy
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Yes, that was essentially what everyone was assuming.
 

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